Chapter 4 - Complex Numbers and Quadratic Equations

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1. Introduction - Imaginary Number i

Real numbers cannot solve x²=-1 because square of any real is ≥0. Invent i=√-1 and i²=-1. x²=-1 → x=±i. Complex number a+ib real part a imaginary part b. Example 3+4i real 3 imag 4.

Power cycle i¹=i i²=-1 i³=-i i⁴=1 repeats every 4. Trick i^n=i^(n mod4). Example i^27 27mod4=3 i³=-i.

Real cannot solve x²=-1 invent i=√-1 i²=-1 x=±i Complex a+ib real a imag b 3+4i real 3 imag 4 Power cycle i -1 -i 1 repeats every 4 i^n=i^(n mod4) i^27 mod4=3 i³=-i

Introduction - Imaginary i

2. Form a+ib and Argand Plane

Standard form z=a+ib a=Re(z) real part b=Im(z) imaginary. Example 3+4i Re=3 Im=4. Argand plane horizontal Real axis Re vertical Imaginary axis Im each complex point (a,b) 3+4i (3,4) 3 right 4 up origin 0+0i=0 distance modulus. Equality a+ib=c+id implies a=c and b=d.

Standard form z=a+ib a=Re(z) b=Im(z) 3+4i Re=3 Im=4 Argand plane horizontal Real Re vertical Imaginary Im each point (a,b) 3+4i (3,4) 3 right 4 up origin 0+0i=0 distance modulus Equality a=c and b=d

Form a+ib Argand Plane

3. Algebra - Addition Subtraction Multiplication Conjugate Modulus

Addition (a+ib)+(c+id)=(a+c)+i(b+d) Example (3+4i)+(2+5i)=5+9i. Subtraction (a-c)+i(b-d) (3+4i)-(2+5i)=1-i. Multiplication (ac-bd)+i(ad+bc) because i²=-1 Example (3+2i)(1+4i)=-5+14i. Conjugate z=a+ib z̄=a-ib mirror real axis 3+4i conjugate 3-4i z+z̄=2a real z·z̄=a²+b²=|z|². Modulus |z|=√(a²+b²) distance origin |3+4i|=5. Division multiply conjugate denominator (a+ib)/(c+id)=(a+ib)(c-id)/(c²+d²).

Addition (a+c)+i(b+d) Example 5+9i Subtraction (a-c)+i(b-d) 1-i Multiplication (ac-bd)+i(ad+bc) i²=-1 Example -5+14i Conjugate z̄=a-ib mirror 3+4i conjugate 3-4i z+z̄=2a z·z̄=|z|² Modulus |z|=√(a²+b²) |3+4i|=5 Division multiply conjugate denominator (c²+d²)

Algebra Conjugate Modulus

4. Polar Form r(cosθ+i sinθ)

Polar form z=a+ib=r(cosθ+i sinθ) r=|z|=√(a²+b²) modulus distance origin θ=arg(z) angle vector origin to (a,b) with positive real axis a=r cosθ b=r sinθ. Argument θ=tan⁻¹(b/a) quadrant adjustment principal (-π,π]. Example z=1+i r=√2 θ=45°=π/4 polar √2(cos45°+i sin45°). Multiplication polar r1r2[cos(θ1+θ2)+i sin(θ1+θ2)] multiply moduli add arguments.

Polar form z=a+ib=r(cosθ+i sinθ) r=|z| modulus distance θ=arg(z) angle vector a=r cosθ b=r sinθ θ=tan⁻¹(b/a) principal (-π,π] Example 1+i r=√2 θ=45°=π/4 polar √2(cos45°+i sin45°) Multiplication polar r1r2[cos(θ1+θ2)+i sin(θ1+θ2)] multiply moduli add arguments

Argand Plane Polar Form

5. Quadratic Equations Complex Roots D<0

Quadratic ax²+bx+c=0 formula x=(-b±√D)/2a D=b²-4ac. D≥0 real roots D<0 complex using i √D=i√|D| Roots x=(-b±i√|D|)/2a conjugate pair. Example x²+2x+5=0 D=-16 √D=4i Roots -1±2i conjugate pair. Property complex roots come conjugate pairs if coefficients real. Nature D>0 two distinct real D=0 one repeated D<0 two complex conjugate no real.

Quadratic ax²+bx+c=0 formula x=(-b±√D)/2a D=b²-4ac D≥0 real D<0 complex i √D=i√|D| Roots (-b±i√|D|)/2a conjugate pair Example x²+2x+5=0 D=-16 Roots -1±2i conjugate pair Property conjugate pairs if coefficients real Nature D>0 two distinct real D=0 one repeated D<0 two complex conjugate

Quadratic Equations Complex Roots

6. Properties Modulus Argument

Conjugate z̄=a-ib z+z̄=2a real z-z̄=2ib imaginary z·z̄=a²+b²=|z|² (z̄)̄=z. Modulus |z|=√(a²+b²)≥0 |z|=0 iff z=0 |z|=|z̄| |z1·z2|=|z1||z2| |z1/z2|=|z1|/|z2| |z1+z2|≤|z1|+|z2| triangle inequality. Argument arg(z1·z2)=arg(z1)+arg(z2) add multiplication arg(z1/z2)=arg(z1)-arg(z2) subtract arg(z̄)=-arg(z) arg(-z)=arg(z)+π.

Conjugate z̄=a-ib z+z̄=2a z-z̄=2ib z·z̄=|z|² (z̄)̄=z Modulus |z|=√(a²+b²) |z|=0 iff z=0 |z|=|z̄| |z1·z2|=|z1||z2| |z1/z2|=|z1|/|z2| |z1+z2|≤|z1|+|z2| triangle Argument arg(z1·z2)=arg(z1)+arg(z2) arg(z1/z2)=arg(z1)-arg(z2) arg(z̄)=-arg(z) arg(-z)=arg(z)+π

Properties Modulus Argument

7. Real Life Applications

Electrical engineering AC circuits impedance Z=R+iX V=I×Z complex Ohm's law. Signals waves sound light radio complex exponential e^(iθ)=cosθ+i sinθ Euler formula Fourier transform break signal into frequencies. Fractals Mandelbrot set iteration z→z²+c beautiful patterns computer graphics. Quantum mechanics Schrödinger wave function ψ complex probability. Control systems aeroplanes rockets stability poles zeros complex plane. GPS mobile MRI imaging complex Fourier transform AC power house electricity.

Electrical AC circuits Z=R+iX V=I×Z Ohm's law Signals waves sound light radio e^(iθ)=cosθ+i sinθ Euler Fourier break signal frequencies Fractals Mandelbrot z→z²+c beautiful patterns graphics Quantum Schrödinger ψ complex probability Control aeroplanes rockets stability poles zeros GPS mobile MRI Fourier AC power house

Real Life Applications

8. Formulas Short Tricks Gold

i=√-1 i²=-1 i³=-i i⁴=1 cycle every 4 i^n=i^(n mod4). z=a+ib Re=a Im=b equality a=c b=d. Addition (a+c)+i(b+d) Subtraction (a-c)+i(b-d) Multiplication (ac-bd)+i(ad+bc) Division multiply conjugate denominator (c²+d²). Conjugate z̄=a-ib z+z̄=2a z-z̄=2ib z·z̄=|z|². Modulus |z|=√(a²+b²) |z|²=z·z̄ |z1·z2|=|z1||z2| |z1+z2|≤|z1|+|z2| triangle. Argument θ=tan⁻¹(b/a) principal (-π,π] Polar r(cosθ+i sinθ) r=|z| θ=arg(z) r1r2[cos(θ1+θ2)+i sin(θ1+θ2)]. Quadratic D=b²-4ac D>0 two distinct real D=0 one repeated D<0 two complex conjugate x=(-b±i√|D|)/2a.

i=√-1 i²=-1 i³=-i i⁴=1 cycle every 4 i^n=i^(n mod4) z=a+ib Re=a Im=b equality a=c b=d Addition (a+c)+i(b+d) Subtraction (a-c)+i(b-d) Multiplication (ac-bd)+i(ad+bc) Division multiply conjugate denominator (c²+d²) Conjugate z̄=a-ib z+z̄=2a z-z̄=2ib z·z̄=|z|² Modulus |z|=√(a²+b²) |z|²=z·z̄ |z1·z2|=|z1||z2| |z1+z2|≤|z1|+|z2| Argument θ=tan⁻¹(b/a) principal (-π,π] Polar r(cosθ+i sinθ) r1r2[cos(θ1+θ2)+i sin(θ1+θ2)] Quadratic D=b²-4ac D>0 two distinct real D=0 one repeated D<0 two complex conjugate x=(-b±i√|D|)/2a

Formulas Short Tricks